Pythagoras number
In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares.
A Pythagorean field is a field with Pythagoras number 1: that is, every sum of squares is already a square.
Examples
- Every non-negative real number is a square, so p = 1.
- For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, so p = 2.
- By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so p = 4.
Properties
- Every positive integer occurs as the Pythagoras number of some formally real field.
- The Pythagoras number is related to the Stufe by p ≤ s + 1. If F is not formally real then s ≤ p ≤ s + 1, and both cases are possible: for F = C we have s = p = 1, whereas for F = F5 we have s = 1, p = 2.
- As a consequence, the Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair of the form or, there exists a field F such that,p) =. For example, quadratically closed fields and fields of characteristic 2 give,p) = ; for primes p ≡ 1, Fp and the p-adic field Qp give ; for primes p ≡ 3, Fp gives, and Qp gives ; Q2 gives, and the function field Q2 gives.
- The Pythagoras number is related to the height of a field F: if F is formally real then h is the smallest power of 2 which is not less than p; if F is not formally real then h = 2s.